Approximation algorithms for guarding holey polygons

Guarding edges of polygons is a version of art gallery problem.The goal is finding the minimum number of guards to cover the edges of a polygon. This problem is NP-hard, and to our knowledge there are approximation algorithms just for simple polygons. In this paper we present two approximation algorithms for guarding polygons with holes.Keywords: guarding, approximation algorithm, vertex guard, edge guar

On rr-Guarding Thin Orthogonal Polygons

Guarding a polygon with few guards is an old and well-studied problem in computational geometry. Here we consider the following variant: We assume that the polygon is orthogonal and thin in some sense, and we consider a point $p$ to guard a point $q$ if and only if the minimum axis-aligned rectangle spanned by $p$ and $q$ is inside the polygon. A simple proof shows that this problem is NP-hard on orthogonal polygons with holes, even if the polygon is thin. If there are no holes, then a thin polygon becomes a tree polygon in the sense that the so-called dual graph of the polygon is a tree. It was known that finding the minimum set of $r$-guards is polynomial for tree polygons, but the run-time was $\tilde{O}(n^{17})$. We show here that with a different approach the running time becomes linear, answering a question posed by Biedl et al. (SoCG 2011). Furthermore, the approach is much more general, allowing to specify subsets of points to guard and guards to use, and it generalizes to polygons with $h$ holes or thickness $K$, becoming fixed-parameter tractable in $h+K$.Comment: 18 page

Visibility Extension via Reflection

This paper studies a variant of the Art Gallery problem in which the "walls" can be replaced by \emph{reflecting-edges}, which allows the guard to see further and thereby see a larger portion of the gallery. We study visibility with specular and diffuse reflections. The number of times a ray can be reflected can be taken as a parameter. The Art Gallery problem has two primary versions: point guarding and vertex guarding. Both versions are proven to be NP-hard by Lee and Aggarwal. We show that several cases of the generalized problem are NP-hard, too. We managed to do this by reducing the 3-SAT and the Subset-Sum problems to the various cases of the generalized problem. We also illustrate that if $\cal P$ is a funnel or a weak visibility polygon, the problem becomes more straightforward and can be solved in polynomial time. We generalize the $\mathcal{O}(\log n)$-approximation ratio algorithm of the vertex guarding problem to work in the presence of reflection. For a bounded $r$, the generalization gives a polynomial-time algorithm with $\mathcal{O}(\log n)$-approximation ratio for several special cases of the generalized problem. Furthermore, Chao Xu proved that although reflection helps the visibility of guards to be expanded, similar to the normal guarding problem, even considering $r$ specular reflections we may need $\lfloor \frac{n}{3} \rfloor$ guards to cover a simple polygon $\cal P$. In this article, we prove that considering $r$ diffuse reflections the minimum number of vertex or boundary guards required to cover $\cal P$ decreases to $\lceil \frac{\alpha}{1+ \lfloor \frac{r}{4} \rfloor} \rceil$, where $\alpha$ indicates the minimum number of guards required to cover $\cal P$ without reflection. funnel or a weak visibility polygon, then the problem becomes more straightforward and can be solved in polynomial time.Comment: 32 pages, 10 figure